Topic: Area of Triangles, Parallelograms, Trapezoids, Kites, Rhombuses, and Composite figures
Strategy: Gallery Walk, Demonstration, Discussion
Level: High School Geometry, Grades 9-11
Situation
This lesson begins a series of lessons dealing with finding areas of different types of polygons. Topically, the
unit ends with students finding the area of composite figures that incorporate regular polygons as well as parts
of circles, parallelograms, rhombuses, kites, trapezoids, and triangles. Students must be able to reason and find
needed dimensions based on properties of the different quadrilaterals as well as by use of trigonometric
relationships and properties of special right triangles. The figure below is the sort of problem students should
be able to answer by the end of the unit.
This lesson in particular is aimed at activating prior knowledge, introducing the notion of decomposition into
simpler shapes and motivating the use of certain formulas, in particular the formula for area of
kites/rhombuses and trapezoids.
Objectives
Recall area formulas for triangles, rectangles and circles.
Use area formula for rhombus and kites.
Reason to find area of polygons by decomposing them into simpler shapes
Use a ruler to measure distances to nearest tenth of a centimeter.
Enduring understandings
1. Area formulas for different types of polygons can be reasoned from an understanding of area for triangles
and parallelograms.
2. Decomposing a composite shape into non-overlapping shapes is a technique for finding the area of the
whole figure.
3. Right triangles and trigonometric relations can assist when direct measurement is not possible when
finding area of figures.
Standards
From Marshfield High School Math Department Course Syllabus on Regular Geometry – 4th Quarter. F.
Perimeter, Area and Volume 1 and 2. “Determine the area of circles, triangles, kites, parallelograms, rectangles,
rhombi, squares, trapezoids, regular polygons, and irregular figures.”
WI Standards addressed – A. Mathematical Processes, C. Geometry, D. Measurement
Procedure
The opening activity is a brainstorm of students' prior knowledge on the topic of area of polygons. On the daily
handout (see attached) Students are prompted to write about what they can recall of area. Projected to the
front of the class are some guiding questions:
1. What is the definition of area?
2. What is area measured in?
3. How do you find the area of a rectangle or square?
4. How do you find the area of a triangle?
5. What do we mean when we say “base” and “height” of a triangle?
6. What about the area of a circle?
After the elapsed time, students are share with a neighbor what they have written down. The teacher solicits
responses along the lines of the guiding questions. Teacher identifies that finding area of triangles and
rectangles are previous skills upon which we will build new understanding in the following lessons.
Next, the teacher leads the class in a demonstration on finding the area of rhombus/kites. Using the interactive
board, the teacher shows how these shapes can be thought of as being made of two congruent triangles. Thus
follows the formula Area = (1/2)(the product of the diagonals). Students take notes on this new material in part
2 of their daily handout (see attached)
The next twenty five minutes are given over to a gallery walk style activity where students must calculate the
area of oversized triangles, circles, rectangles, parallelograms, semicircles, trapezoids, etc. around the room.
Students are paired by finding the circled number on their daily handout (teacher circles these before start of
class). There are ten shapes around the classroom. Students must use their class formula sheet, notes, and
meter stick to approximate areas. Teacher briefs proper work flow and expectations before releasing pairs to
start the activity.
During the activity, the teacher circulates, probing students to identify the correct formulas to use, and in the
case of the pentagon, semicircle and trapezoid, the notion of decomposing the shape into simpler shapes,
calculating these areas and summing or subtracting them to find total area.
Following the activity, the teacher leads a discussion on results, posts an answer key for comparison, recounts
various techniques tried by different groups to find areas of more difficult shapes and discusses sources of
error.
The teacher closes class by assigning the daily homework assignment (listed on the bottom of the handout, see
attached) and instructing students that they will in the course of their homework need to use the Pythagorean
Theorem, Properties of special right triangles, and trigonometric relations.
Materials
Student handouts
10 Meter sticks
10 oversized figures
Calculators
Smart Board
Geometer's Sketchpad or ActiveInspire software
Assessment
Formative
Students compare their results with the posted answer key, factoring in sources of error. The discussion of
various techniques, both flawed or correct serve to give feedback to the student on whether they correctly
found the areas of the various figures.
The daily homework assignment is likewise formative – students can check answers to odd numbered
problems.
Summative
Students complete and turn in an assignment after the first three lessons. Page 1 of the assignment assesses
this particular lesson's objectives:
Furthermore, an in class quiz further assessed student mastery of the objective. Attached is a form of the quiz.
Differentiation and Addressing Learning Styles
The lesson design provides for multiple approaches to the content. Students both individually and with pairs
brainstorm, measure, reason, debate, discuss and evaluate their thinking in solving for area. The social nature
of the gallery walk plays to the strengths of interpersonal learners. The demonstration involving the
decomposition of a kite into two congruent triangles and derivation of the formula provides for visual and
spatial-logic learners. The manipulation of the oversized figures and use of the meter stick provides for
kinesthetic learners. Discussion and teacher led questioning follows the auditory track. Students get practice
applying the skill in multiple formats.
Lesson Plan Reflection
The key to this lesson was activation of prior knowledge. Students have had objectives related to area
of triangles, rectangles and circles as early as fifth grade. Trapezoids and parallelograms are introduced later.
What is new to this treatment of area is the introduction of trigonometry to find needed measures that are
missing, the kites/rhombus formula, and finally using decomposition into simpler shapes to find the area of
regular polygons and composite figures. In deciding which activity to use, I wanted to steer clear of anything
just introducing formula and setting the students loose on problems. Although using formula and substitutions
is a necessary skill that the students still need practice with, it is repetitive, low level application and does not
usually lend itself to retention. Also the pairing and “decentralized” aspect of this particular lessoned gave large
allowance to variance in student’s readiness and level of mastery with this topic.
Also, this lesson truly did allow me to access and honor many different approaches to learning – visualspatial, auditory, interpersonal, kinesthetic, logical-mathematics are all given about equal weight in this lesson,
which is usually not possible to accomplish in the span of one lesson. Although the students did require quite a
bit of coaxing and probing during the activity to venture an attempt, their questions were forthcoming and
their misunderstandings were made apparent in a non-threatening way since it was not in front of the whole
class. This is helpful for students who have difficulties participating in class because they fear publically being
seen as behind or “slow.”
The in class quiz that was used after this lesson showed that about 3 out of every 4 students had really
mastered the objectives introduced here. The newest material dealing with kites and rhombuses showed the
most inconsistent results. I think there may have been a good deal of proactive interference with the “older”
knowledge dealing with area in rectangles and parallelograms. I have therefore included extra practice time for
rhombus and kite problems in the review prior to our summative unit test.
There was some excellent incidental things that came up in the course of the lesson that was beneficial
if unplanned for. First was the common misconception that since 100 cm = 1 m that 100 square centimeters = 1
square meter. Also that notion that there are no fewer than three base/height pairs in any triangle and lastly
that a rhombus is a special case of a parallelogram, so that it is possible to make use of two different formulas,
depending on what the given information is. In future iterations of this lesson, it would be helpful to have a few
guiding questions dealing with these things so that all students consider them.
Student Materials